标题： 度量图上漂移扩散方程的PINN方法比较 显示英文标题

标题： A comparison of PINN approaches for drift-diffusion equations on metric graphs

摘要： 在本文中，我们专注于比较用于量子图的机器学习方法，量子图是度量图，即具有专用边长的图，并且有一个相关的微分算子。 在我们的案例中，微分方程是一个漂移-扩散模型。 量子图的计算方法需要对微分算子进行仔细的离散化，同时还要包含节点条件，在我们的案例中是基尔霍夫-诺伊曼条件。 传统的数值方案已经相当成熟，但当微分方程成为优化问题中的约束时，必须手动调整。 最近，物理信息神经网络（PINNs）作为一种多功能工具，已出现用于解决来自多个应用的偏微分方程。 它们通过仅略微改变用于正向模拟的问题公式，就能灵活地解决参数识别或优化问题。 我们比较了多种PINN方法来解决度量图上的漂移-扩散问题。 显示英文摘要

摘要： In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.